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Homomorphisms of the algebra of locally integrable functions on the half line

Part of: Abstract harmonic analysis Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  09 April 2009

Sandy Grabiner
Sandy Grabiner
Affiliation:
Department of Mathematics, Pomona College, 610 North College Avenue, Claremont, CA 91711, USA e-mail: [email protected]
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Abstract

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Let φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.

MSC classification

Secondary: 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 46J40: Structure, classification of commutative topological algebras 46J45: Radical Banach algebras 43A20: $L^1$-algebras on groups, semigroups, etc. 43A10: Measure algebras on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 253 - 278
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Bade, W. G. and Dales, H. G., ‘Continuity of derivations from radical convolution algebras’, Studia Math. 95 (1989), 60–91.CrossRefGoogle Scholar
[2]Dales, H. G., Banach algebras and automatic continuity, London Math. Soc. Monographs 24 (Clarendon Press, Oxford, 2000).Google Scholar
[3]Doran, R. S. and Wichman, J., Approximate identities and factorization in Banach modules, Lecture Notes in Math. 768 (Springer, Berlin, 1979).CrossRefGoogle Scholar
[4]Dunford, N. and Schwartz, J. T., Linear operators. Part I (Wiley Interscience, New York, 1958).Google Scholar
[5]Ghahramani, F., ‘Homomorphisms and derivations on weighted convolution algebras’, J. London Math. Soc. (2) 21 (1980), 149–161.Google Scholar
[6]Ghahramani, F., ‘Isomorphisms between radical weighted convolution algebras’, Proc. Edin. Math. Soc. (2) 26 (1983), 343–351.CrossRefGoogle Scholar
[7]Ghahramani, F. and Grabiner, S., ‘Standard homomorphisms and convergent sequences in weighted convolution algebras’, Illinois J. Math. 36 (1992), 505–527.CrossRefGoogle Scholar
[8]Ghahramani, F. and Grabiner, S., ‘The Lp theory of standard homomorphisms’, Pacific J. Math. 168 (1995), 49–60.CrossRefGoogle Scholar
[9]Ghahramani, F., Grabiner, S. and McClure, J. P., ‘Standard homomorphisms and regulated weights on weighted convolution algebras’, J. Funct. Anal. 91 (1990), 278–286.CrossRefGoogle Scholar
[10]Ghahramani, F. and McClure, J. P., ‘Automorphisms and derivations of a Fréchet algebra of locally integrable functions’, Studia Math. 103 (1992), 51–69.Google Scholar
[11]Grabiner, S., ‘Weighted convolution algebras on the half-line’, J. Math. Anal. Appl. 83 (1981), 531–553.CrossRefGoogle Scholar
[12]Grabiner, S., ‘Extremely non-standard ideals and non-injective operational calculi’, J. London Math. Soc. (2) 30 (1984), 129–135.CrossRefGoogle Scholar
[13]Grabiner, S., ‘Homomorphisms and semigroups in weighted convolution algebras’, Indiana Univ. Math. J. 37 (1988), 589–615.CrossRefGoogle Scholar
[14]Grabiner, S., ‘Semigroups and the structure of weighted convolution algebras’, in: Proceedings of the conference on automatic continuity and Banach algebras (ed. Loy, R. J.), Proc. Centre Math. Anal. Austral. Nat. Univ. 21 (1989) pp. 155–169.Google Scholar
[15]Grabiner, S., ‘Weighted convolution algebras and their homomorphisms’, in: Functional Analysis and Operator theory, Banach Center Publ. 30 (Polish Acad. Sci., Warsaw, 1994) pp. 175–190.Google Scholar
[16]Grabiner, S., ‘Weak* properties of weighted convolution algebras’, Proc. Amer. Math. Soc. 132 (2004), 1675–1684.CrossRefGoogle Scholar
[17]Gripenberg, G., Londen, S.-O. and Staffans, O., Volterra integral and functional equations, Encyclopedia Math. Appl. 34 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[18]Hille, E. and Phillips, R. S., Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31 (Amer. Math. Soc, Providence, RI, 1957).Google Scholar
[19]Kamowitz, H. and Scheinberg, S., ‘Derivations and automorphisms of L 1 (0,1)’, Trans. Amer.Math.Soc. 135 (1969), 415–427.Google Scholar